Both are full-featured free and open source web applications with which you can realize a very diverse variety of projects.
Both are written in PHP and use a database such as MySQL
Both can run on standard, inexpensive hosting.
Both have large & vibrant communities
Both have a template layer
Tiki is more centralized. Tiki is the all-in-one model - all modules are part of the package while Typo3 (like Joomla!) is the small-core-and-add-what-you-need model (you need to use third party modules out of the package). Each model has its pros & cons.
Tiki is more focused on collaboration and structured wiki (customisable data management) over the web while Typo3 is more focused on web publishing.